Math 166Z Homework # 11
Due Thursday April 28th
1. Approximate each function below with the nth order Taylor polynomial Pn (x) centered
at the number a. Then estimate the accuracy of your approximation for x in the given
interval.
π
(a) f (x) = sin x, a = π4 , n = 5, 0 ≤ x ≤
2
(b) f (x) = ln x, a = 4, n = 3, 3 ≤ x ≤ 5
2. For each set of parametric equations below:
(1) Sketch the curve
(2) Eliminate the parameter to find the Cartesian equation of the curve
dy
d2 y
(2) Find
and
dx
dx
√
(a) x = t, y = 1 − t
(b) x = sin θ, y = cos θ, 0 ≤ θ ≤ π
(c) x = 3t2 , y = 2 + 5t, 0 ≤ t ≤ 2
3. Find the slope of the tangent line to the given polar curve at the point specified by the
value of θ.
π
(a) r = cos θ + sin θ, θ =
4
(b) r = ln θ, θ = e
4. Find the area of the region that is bounded by the given curve and lies in the given
sector.
π
(a) r = 2 cos θ, 0 ≤ θ ≤
2
π
θ
(b) r = e , 0 ≤ θ ≤
6
1
5π
(c) r = , 0 ≤ θ ≤
θ
6
5. Find the area of the region enclosed by one loop of the curve.
(a) r = sin 3θ
(b) r = 2 cos 4θ